MinT - Best Known (23, 23+23, s)-Nets in Base 25

Best Known (23, 23+23, s)-Nets in Base 25

(23, 23+23, 156)-Net over F₂₅ — Constructive and digital

Digital (23, 46, 156)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 14, 52)-net over F₂₅, using
  - net from sequence [i] based on digital (3, 51)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 52, using
      - a function field by GarcÃa and Quoos [i]
2. digital (9, 32, 104)-net over F₂₅, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by GarcÃa and Quoos [i]

(23, 23+23, 349)-Net over F₂₅ — Digital

Digital (23, 46, 349)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁴⁶, 349, F₂₅, 23) (dual of [349, 303, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(25⁴⁶, 624, F₂₅, 23) (dual of [624, 578, 24]-code), using
  - the primitive BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â {3,4,…,25}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(23, 23+23, 107059)-Net in Base 25 — Upper bound on s

There is no (23, 46, 107060)-net in base 25, because

1 times m-reduction [i] would yield (23, 45, 107060)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 807 813530 609496 388224 943791 723434 503499 211750 878094 910786 108705 > 25⁴⁵ [i]